To quantitatively describe this device, I start with the assumption that the medium is an ensemble of two-level atoms with ground state $a$ and exited state $n$ separated by energy difference $\hbar \omega_{na}$. In this case, the medium can be described by a frequency-dependent absorption coefficient given by
$$\alpha(\omega) = \frac{4\pi N}{\hbar c} \frac{\omega_{na}|\mu_{na}|^2}{\gamma_{na}} \left[\frac{\gamma_{na}^2}{(\omega_{na} - \omega)^2+\gamma_{na}^2}\right]$$
where $N$ is the atomic number density, $\gamma_{na}$ is the dipole dephasing rate\footnote{$\gamma_{na}$ is also the half width at half maximum in angular frequency units of the absorption line, in the limit of weak fields}, and $\mu_{na}$ is the dipole matrix element for the $n\rightarrow a$ transition. This expression can be derived from a density matrix calculation of the linear susceptibility by assuming an isotropic medium \cite{Boyd_2002aa}.

In order to quantify the individual contribution of each atom to the absorption, and thus to the action of this all-optical switch, it is useful to consider the absorption cross section $\sigma$, defined in terms of the absorption as
$$\sigma=\alpha/N.$$
The cross section can hence be interpreted as the effective area of an atom for absorbing radiation from an incident beam of light. Next, I compute the absorption cross section for this prototypical switch by making a few simplifying assumptions. First, by the definition,
$$\sigma = \frac{4\pi}{\hbar c} \frac{\omega_{na}|\mu_{na}|^2}{\gamma_{na}} \left[\frac{\gamma_{na}^2}{(\omega_{na} - \omega)^2+\gamma_{na}^2}\right]$$
which for the case of resonant light $\omega=\omega_{na}$ becomes
\begin{equation}
  \label{eqn:sigma_res}
  \sigma = \frac{4\pi}{\hbar c} \frac{\omega_{na}|\mu_{na}|^2}{\gamma_{na}}.
\end{equation}

The first assumption involves the dipole dephasing rate $\gamma_{na}$ which can be represented as
\begin{equation}
  \label{eqn:gamma}
  \gamma_{na}=\frac{1}{2} \left(\Gamma_n+\Gamma_a\right) + \gamma_{na}^{\mathrm{col}},
\end{equation}
where $\Gamma_n$ and $\Gamma_a$ are the total decay rates of population out of the $n$ and $a$ levels, respectively. The quantity $\gamma_{na}^{\mathrm{col}}$ is the dipole dephasing rate due to processes that are not associated with the transfer of population, such as elastic collisions.

From Eq.~\ref{eqn:sigma_res}, note that the absorption cross section will have its maximum value when $\gamma_{na}^{\mathrm{col}}=0$. Assuming $\gamma_{na}^{\mathrm{col}}=0$, \emph{i.e.} there is no dephasing due to collisions, then Eq.~\ref{eqn:gamma} will be minimized. Using a further assumption, that $a$ is a ground state, then its decay rate $\Gamma_a$ must vanish, leaving the minimum possible value of $\gamma_{na}$ to be $\frac{1}{2}\Gamma_n$.

If the state $n$ can decay only to the ground state $a$, then the decay rate is simply given by the Einstein A coefficient
$$\Gamma_n=\frac{4\omega_{na}^3|\mu_{na}|^2}{3\hbar c^3}.$$ Using $\gamma_{na}=\frac{1}{2}\Gamma_n$ in Eq.~\ref{eqn:sigma_res}, I find that the maximum value of the absorption cross section is
$$\sigma_\mathrm{max}=\frac{3\lambda^2}{2\pi},$$ making use of the relation $c/\omega=\lambda/(2\pi)$.

All of the tools are now in place to evaluate the behavior of this switch for realistic parameters. First, since I have assumed that there are no broadening effects aside from natural broadening, an appropriate system for this approximation is a sample of cold atoms such as the cloud of atoms in a magneto-optical trap. Consider a trapped ensemble of Rubidium atoms with resonant wavelength $\lambda=780$~nm, trap diameter $L=2$~mm, and atomic number density of $10^{10}$~cm$^{-1}$. In this medium, a saturating control beam of intensity $I_s=(\hbar \omega)(\sigma \tau)=10^{-2}$~W/cm$^2$ causes the absorption experienced by a weak signal beam, $I<<I_s$, to decrease, as shown in Fig.~\ref{fig:cold_absorption}.
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/cold_abs.pdf}
  \end{center}
  \caption[Reduced absorption with a saturating beam for cold, natural-linewidth, Rubidium vapor.]{Reduced absorption with a saturating beam for cold, naturally broadened, Rubidium vapor. The dashed red line indicates the intensity as a function of position in the medium, $z$, for the case of no saturating beam. When the saturating beam is present, the absorption is reduced, and transmission increases, as shown by the solid blue curve.}
  \label{fig:cold_absorption}
\end{figure}
After traversing the medium, the signal propagating through an unsaturated medium has a transmission of 14\%. With the control beam applied, and thus saturating the medium, the transmission increases to 38\%. 

where I have approximated $\theta$ to second order as a small angle. By considering the extra length of the off-axis wavevector due to weak-wave retardation I can replace $k_w=k_s+\Delta k$
\begin{equation}
  1-\frac{\theta^2}{2}\simeq\frac{k_s}{k_s+\Delta k}=\frac{1}{1+\frac{\Delta k}{k_s}}
\end{equation}
by assuming the quantity $\Delta k/k_s$ is small,
\begin{align}
  \frac{\theta^2}{2}&\simeq\frac{\Delta k}{k_s}\\
  \theta&=\sqrt{\frac{2\Delta k}{k_s}}=\sqrt{\frac{2\Delta n}{n_s}},
\end{align}
